code: def test_basis_rotation_utransform(core, one_electron, two_electron):
      r"""Test that basis_rotation function returns the correct transformation matrices. This test
      constructs the matrix representation of a factorized Hamiltonian and then applies the
      transformation matrices to generate a new set of fermionic creation and annihilation operators.
      A new Hamiltonian is generated from these operators and is compared with the original
      Hamiltonian.
      """
      *_, u_transform = qml.qchem.basis_rotation(
          one_electron, two_electron
      )
  
      a_cr = [  # fermionic creation operators
          np.array(
              [
                  [0.0, 0.0, 0.0, 0.0],
                  [0.0, 0.0, 0.0, 0.0],
                  [1.0, 0.0, 0.0, 0.0],
                  [0.0, 1.0, 0.0, 0.0],
              ]
          ),
          np.array(
              [
                  [0.0, 0.0, 0.0, 0.0],
                  [1.0, 0.0, 0.0, 0.0],
                  [0.0, 0.0, 0.0, 0.0],
                  [0.0, 0.0, 1.0, 0.0],
              ]
          ),
      ]
  
      a_an = [  # fermionic annihilation operators
          np.array(
              [
                  [0.0, 0.0, 1.0, 0.0],
                  [0.0, 0.0, 0.0, 1.0],
                  [0.0, 0.0, 0.0, 0.0],
                  [0.0, 0.0, 0.0, 0.0],
              ]
          ),
          np.array(
              [
                  [0.0, 1.0, 0.0, 0.0],
                  [0.0, 0.0, 0.0, 0.0],
                  [0.0, 0.0, 0.0, 1.0],
                  [0.0, 0.0, 0.0, 0.0],
              ]
          ),
      ]
  
      # compute matrix representation of the factorized Hamiltonian. Only the one-body part is
      # included to keep the test simple.
  
      t_matrix = one_electron - 0.5 * np.einsum("illj", two_electron)
  
      h1 = 0.0
      for i, ac in enumerate(a_cr):
          for j, aa in enumerate(a_an):
              h1 += t_matrix[i, j] * ac @ aa
  
      h1 += np.identity(len(h1)) * core
  
      # compute matrix representation of the basis-rotated Hamiltonian. Only the one-body part is
      # included to keep the test simple.
  
      u = u_transform[0]
      ac_rot = []
      aa_rot = []
  
      for u_ in u.T:  # construct the rotated creation operators
          ac_new = 0.0
          for i, ac in enumerate(a_cr):
              ac_new += u_[i] * ac
          ac_rot.append(ac_new)
  
      for u_ in u.T:  # construct the rotated annihilation operators
          aa_new = 0.0
          for i, aa in enumerate(a_an):
              aa_new += u_[i] * aa
          aa_rot.append(aa_new)
  
      val, _ = np.linalg.eigh(t_matrix)
      h2 = 0.0
      for i, v in enumerate(val):
          h2 += v * ac_rot[i] @ aa_rot[i]
  
      h2 += np.identity(len(h2)) * core
  
      assert np.allclose(h1, h2)
  
file_path: pennylane/tests/qchem/test_factorization.py
imports: import pytest
  import pennylane
  from pennylane import numpy
label_desc: Use image node_id 6 for calling a global function with example usage: test_basis_rotation_utransform(core, one_electron, two_electron) without return types